Find point M (vectors)

I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.


Help anyone?

Quote:

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Originally Posted by OReilly

I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.


Help anyone?

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I guess that is the center (where diagnols intersect).
Since that is the only which makes sense.
Check to see if that is true.

Quote:

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Originally Posted by ThePerfectHacker

I guess that is the center (where diagnols intersect).
Since that is the only which makes sense.
Check to see if that is true.

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Well, since diagonals are not always equal I can't say that is the right point.

If it were equal then it could be MC=1/2AC, MA=-1/2AC, MD = 1/2BD, MB = -1/2BD and we would have two pairs of opposite vectors.

Quote:

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Originally Posted by OReilly

I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.
Help anyone?

---

Hello, OReilly,

I've attached an image which show you what to do in 4 steps. The steps are numbered.

1. Construct the midpoint of AD, call it P.
2. Construct the midpoint of BC, call it Q.
3. Construct the midpoint of PQ: That's M.

If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.

In step #4 you see the complete result.

Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0

EB

Quote:

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Originally Posted by earboth

Hello, OReilly,

I've attached an image which show you what to do in 4 steps. The steps are numbered.

1. Construct the midpoint of AD, call it P.
2. Construct the midpoint of BC, call it Q.
3. Construct the midpoint of PQ: That's M.

If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.

In step #4 you see the complete result.

Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0

EB

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Thanks, very nice solution.

Hello, OReilly,

if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.

If
A(a1 , a2)
B(b1 , b2)
C(c1 , c2)
D(d1 , d2)

then

M(¼ (a1+b1+c1+d1), ¼ (a2+b2+c2+d2))

EB

Quote:

---

Originally Posted by earboth

Hello, OReilly,

if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.

If
A(a1 , a2)
B(b1 , b2)
C(c1 , c2)
D(d1 , d2)

then

M(¼ (a1+b1+c1+d1), ¼ (a2+b2+c2+d2))

EB

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Can this point be actually found in this way:

It is the point of intersection of lines that are dividing two pairs of opposite sides of quadrilateral in half.